As a surprising effect, in the systems where the noninteracting
Brownian bugs undergo death and reproduction with equal
probabilities, spatial clustering of the bugs was observed in
single realizations [1]}, [2]}, [3]}.
A typical time evolution of such a system is illustrated in
Fig. REF a. We note that in all figures presenting
the spatial configurations of the bugs, we have divided the
individuals according to their initial position into nine
groups characterized by different colors as in
Fig. REF  at time \(t = 0\) ; if reproduction takes
place, the newborn bug assumes the same color as the parent.
From Fig. REF a one can see that many small clusters
form some time after starting from a uniform initial
distribution. The occurrence of the clustering is related to
the fact that in the case of reproduction the new bug is
located at the same position as the parent. Due to the
fluctuations and irreversibility of death the number of
clusters decreases in time, until there will be a single
cluster consisting of individuals coming from a single
ancestor. There are constant, spontaneous, short-lived
break-offs from the main cluster, which are always located near
it. The center of mass of such a cluster undergoes a motion
similar to that of a single bug [1]} and its linear
width fluctuates with a typical value proportional to
\(\sqrt{N_0}\)  [1]}. Furthermore, the larger the
diffusion coefficient \(\kappa \) , the wider is the cluster
(notice that when simulating the system numerically, if the
diffusivity becomes so large that the jump lengths become
comparable to the system size, one needs to take a larger
simulation box). Finally, due to the fluctuations also the last
cluster disappears.
